# Structural Break Clarification

## Sup-F Scan (All Breakpoints)

| Break Year | F-stat | p-value | Sig | Z₁ (pre) | Z₁ (post) |
|---|---|---|---|---|---|
| 1999 | 5.78 | 0.0000 | *** | 36.80 | 16.77 |
| 2000 | 11.37 | 0.0000 | *** | 25.55 | 15.19 |
| 2001 | 20.30 | 0.0000 | *** | 39.88 | 15.88 |
| 2002 | 14.36 | 0.0000 | *** | 17.52 | 16.52 |
| 2003 | 14.57 | 0.0000 | *** | 7.45 | 15.82 |
| 2004 | 8.84 | 0.0000 | *** | 9.51 | 16.87 |
| 2005 | 15.33 | 0.0000 | *** | 3.69 | 16.09 |
| 2006 | 16.21 | 0.0000 | *** | 10.78 | 18.88 |
| 2007 | 16.78 | 0.0000 | *** | 8.76 | 20.15 |
| 2008 | 16.07 | 0.0000 | *** | 9.49 | 19.96 |
| 2009 | 12.46 | 0.0000 | *** | 10.92 | 19.05 |
| 2010 | 11.98 | 0.0000 | *** | 14.97 | 18.28 |
| 2011 | -5.86 | 1.0000 |  | 11.74 | 19.09 |
| 2012 | -7.64 | 1.0000 |  | 13.72 | 13.79 |
| 2013 | 0.13 | 0.9982 |  | 16.85 | 14.83 |
| 2014 | 3.44 | 0.0006 | *** | 16.56 | 16.33 |
| 2015 | 2.68 | 0.0062 | *** | 14.96 | 17.83 |
| 2016 | 2.60 | 0.0079 | *** | 13.62 | 16.81 |

**Sup-F breakpoint**: 2001 (F=20.30, p=0.0000)

## Specific Chow Tests

| Break | F-stat | p-value | Sig |
|---|---|---|---|
| 2001 | 20.30 | 0.0000 | *** |
| 2008 | 16.07 | 0.0000 | *** |
| 2001 (Sup-F) | 20.30 | 0.0000 | *** |

## Interaction Approach (Z₁ × post dummy)

| Break | Z₁ | Sig | Post | Sig | Z₁×Post | Sig | R² | N |
|---|---|---|---|---|---|---|---|---|
| 2001 | 16.4591 | *** | -0.1016 |  | -0.0276 |  | 0.498 | 1,868 |
| 2008 | 16.2870 | *** | -0.0470 |  | -0.0455 |  | 0.497 | 1,868 |
| 2001 | 16.4591 | *** | -0.1016 |  | -0.0276 |  | 0.498 | 1,868 |

## Interpretation

The Sup-F test detects the breakpoint with **maximum** instability across all candidate years. The Chow test at a specific date (e.g. 2008) tests only that single break. The interaction approach tests whether Z₁'s effect changes **within** a single regression, but has lower power than the split-sample Chow test because it imposes a common error variance across regimes.

When the Sup-F detects instability but the Z₁×post interaction is insignificant, this typically means the break affects **multiple** coefficients (not just Z₁), or the structural change is gradual rather than a sharp regime switch.